239,500,800
12!/2!
* * * * *
Actually, as the word "permutation" [not permutations] has 11 letters, the answer is 11!/2! = 19,958,400
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In permutations the order of the elements does matter. In combinations it does not.So, the permutations 1,2,3 and 3,1,2 are not the same. But they are the same combination.
10! permutations of the word "Arithmetic" may be made.
If the first and second C are indistinguishable, then there are 554,400 permutations. If one is upper case and the other is lower case, then there are twice as many.
The number of permutations of the letters MATHEMATICS is 11 factorial or 39,916,800. However, since the M is repeated once you must divide by 2. You must also divide by 2 because the A is repeated once. And again for the T. This results in an overall distinct permutation count of 4,989,600.
There are 7 factorial, or 5040 permutations of the letters in the word NUMBERS.